Finite Group Databasebeta

← #28373494

Subgroups — #28373494

Order 13952.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812732C211yesmaximal, characteristic, complemented
2C2×#16812729C211yesmaximal, characteristic, complemented
3C2×#16812725C211yesmaximal, characteristic, complemented
4#20295572C211yesmaximal, characteristic, complemented
5C2×#16812686C211yesmaximal, characteristic, complemented
6#20295417C212yesmaximal, complemented
7#20295414C212yesmaximal, complemented
8#20295409C212yesmaximal, complemented
9#20295407C212yesmaximal, complemented
10M16×Dic109C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×Dic218C2211yescharacteristic
14#16812732—24no—
15#16812729—24no—
16#16812725—24no—
17C2×Dic436C2212yescomplemented
18C2×#13182019C2212yescomplemented
19C2×#13182018C2212yescomplemented
20C2×#13182017C2212yes—
21C2×#13182017C2212yescomplemented
22C2×#13182014C2212yes—
23C2×#13182013C2211yescharacteristic
24#16812697C2211yescharacteristic, complemented
25#16812695C2211yescharacteristic, complemented
26#16812688C2211yescharacteristic, complemented
27#16812686—22no—
28#16812686C2212yes—
29#16812686C2212yes—
30#16812686C2211yescharacteristic, complemented
31#16812671—22no—
32C4×C22⋊D109—22no—
33C2×C22⋊Dic109—22no—
34C2×C4×Dic109C2211yescharacteristic
35#16812609C2212yescomplemented
36C8×Dic109C2212yescomplemented
37C2×#41×C109C2211yescharacteristic
38C22×D4×C109C2211yescharacteristic
39AGL(1,ℤ/8)×C109—24no—
40C2×SD16×C109C2212yes—
41C2×D8×C109C2212yes—
42C2×M16×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218—22no—
46C2×Dic218—22no—
47C2×Dic218—21no—
48C2×Dic218C2312yes—
49C2×Dic218C2311yescharacteristic
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—21no—
52C2×C22⋊D109—22no—
53C22×Dic109—21no—
54C22×Dic109D412yes—
55C2×C4×D109—21no—
56Dic436—24no—
57#13182019—24no—
58#13182018—24no—
59#13182017—24no—
60#13182017—24no—
61#13182014—24no—
62#13182013D414yescomplemented
63C2×#11724784C2312yes—
64#13182008—22no—
65C22⋊Dic109—24no—
66C22⋊Dic109—24no—
67C22⋊Dic109—22no—
68#13182006—22no—
69C4×Dic109D412yes—
70C4×Dic109C2311yescharacteristic
71C4×Dic109C2311yescharacteristic
72C24×C109—21no—
73#41×C109—22no—
74#41×C109—22no—
75C2×Q8×C109C2311yescharacteristic
76C2×D4×C109—21no—
77C2×D4×C109—22no—
78C2×D4×C109—22no—
79C2×D4×C109—21no—
80C2×D4×C109C2311yescharacteristic
81C2×D4×C109C2312yes—
82C22×C4×C109—21no—
83C22×C4×C109C2311yescharacteristic
84SD16×C109—24no—
85D8×C109—24no—
86M16×C109D414yescomplemented
87C2×C8×C109C2312yes—
88C23×C109—21no—
89C23×C109—22no—
90C23×C109—44no—
91C23×C109—42no—
92C23×C109C2×D411yescharacteristic
93C22×D109—21no—
94Q8×C109—21no—
95Q8×C109—21no—
96D4×C109—21no—
97D4×C109—22no—
98D4×C109—22no—
99D4×C109—21no—
100D4×C109—42no—
101D4×C109—42no—
102C2×C4×C109—21no—
103C2×C4×C109—42no—
104C2×C4×C109C2×D411yescharacteristic
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2×D412yes—
107C2×C4×C109C2411yescharacteristic
108C22⋊D109—44no—
109C22⋊D109—42no—
110C2×Dic109—22no—
111C2×Dic109—21no—
112C2×Dic109—42no—
113C2×Dic109—42no—
114C2×Dic109C2×D412yes—
115C2×Dic109C2×D412yes—
116D436—21no—
117D436—21no—
118C4×D109—42no—
119Dic218—21no—
120Dic218—22no—
121Dic218—22no—
122Dic218—21no—
123Dic218—42no—
124C8×C109C2×D414yes—
125#11724784C2×D414yes—
126C22×C109—22no—
127C22×C109—41no—
128C22×C109—42no—
129C22×C109—42no—
130C22×C109—44no—
131C22×C109—84no—
132C22×C109#12712yes—
133C22×C109C22×D411yescharacteristic
134C2×D109—41no—
135C2×D109—42no—
136C4×C109—41no—
137C4×C109#12712yes—
138C4×C109C22×D411yescharacteristic
139C4×C109C22×D411yescharacteristic
140Dic109—22no—
141Dic109—22no—
142Dic109—42no—
143Dic109—41no—
144C2×C109—22no—
145C2×C109—81no—
146C2×C109—82no—
147C2×C109C2×#12711yescharacteristic
148C2×C109#57512yes—
149D109—81no—
150#3153—1091nomaximal, Sylow
151C109#315311yescharacteristic, complemented, Sylow
152C2×#137—1091no—
153C2×#137—1091no—
154C2×AGL(1,ℤ/8)—1091no—
155C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
156#532—1091no—
157C2×#124—1091no—
158#497—1092no—
159#496—1092no—
160#495—1092no—
161#495—1092no—
162C4×M16—1091no—
163C2×#41—1091no—
164C2×#41C2×D10911yescharacteristic
165C22×Q8—1091no—
166C22×D4C2×D10911yescharacteristic
167#137—2184no—
168#137—2184no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2184no—
171C2×Q16—1092no—
172C2×Q16—1092no—
173C2×SD16—1092no—
174C2×SD16—1092no—
175C2×SD16—1092no—
176C2×SD16C2×D10912yes—
177C2×D8—1092no—
178C2×D8C2×D10912yes—
179C2×M16—1091no—
180C2×M16C2×D10911yescharacteristic
181C4⋊Q8—1091no—
182#127—1091no—
183#124—1091no—
184#124—1092no—
185#124—1092no—
186#124—1091no—
187#124—2182no—
188#121—2182no—
189C4×D4—2182no—
190C2×C22⋊C4—2182no—
191C2×C42—1091no—
192#97—1092no—
193C4×C8—1092no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2182no—
198#41—2182no—
199C2×Q8—1092no—
200C2×Q8—1091no—
201C2×Q8—2182no—
202C2×Q8—2182no—
203C2×Q8—2181no—
204C2×Q8C22×D10911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1091no—
210C2×D4—2182no—
211C2×D4—2181no—
212C2×D4C22×D10912yes—
213C2×D4C22×D10911yescharacteristic
214C22×C4—21no—
215C22×C4—1092no—
216C22×C4—2181no—
217C22×C4—2181no—
218C22×C4C22×D10911yescharacteristic
219Q16—2184no—
220Q16—2184no—
221SD16—24no—
222SD16—2184no—
223SD16—2184no—
224SD16—2184no—
225D8—24no—
226D8—2184no—
227M16—1094no—
228M16C22⋊D10914yescomplemented
229C2×C8—1092no—
230C2×C8C22×D10912yes—
231C4⋊C4—2182no—
232C22⋊C4—2182no—
233C22⋊C4—2184no—
234C22⋊C4—2184no—
235C22⋊C4—2182no—
236C42—1092no—
237C42—1091no—
238C42—1091no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2181no—
244C23D4×D10911yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—4362no—
248Q8—2181no—
249Q8—2182no—
250Q8—2182no—
251Q8—2181no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—4362no—
259D4—4364no—
260D4—2181no—
261D4—2181no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1092no—
265C2×C4—1092no—
266C2×C4—4362no—
267C2×C4—4362no—
268C2×C4—4362no—
269C2×C4—2182no—
270C2×C4—2181no—
271C2×C4D4×D10911yescharacteristic
272C2×C4D4×D10912yes—
273C2×C4C2×C22⋊D10912yes—
274C2×C4C23×D10911yescharacteristic
275C8—1094no—
276C8C2×C22⋊D10914yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—4361no—
284C22—4362no—
285C22#1681269512yes—
286C22C2×D4×D10911yescharacteristic
287C4—41no—
288C4—4361no—
289C4—4362no—
290C4—2182no—
291C4—2182no—
292C4#1681269512yes—
293C4C2×D4×D10911yescharacteristic
294C4C22×C22⋊D10911yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—8721no—
299C2C2×#1681269511yescharacteristic
300C2#2029582212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3153
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×#16812729C2semidirect
3. C2×#16812725C2semidirect
4. #20295572C2semidirect
5. C2×#16812686C2semidirect
6. #20295417C2semidirect
7. #20295414C2semidirect
8. #20295409C2semidirect
9. #20295407C2semidirect
10. M16×Dic109C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
17. C2×Dic436C22semidirect
18. C2×#13182019C22semidirect
19. C2×#13182018C22semidirect
21. C2×#13182017C22semidirect
24. #16812697C22semidirect
25. #16812695C22semidirect
26. #16812688C22semidirect
30. #16812686C22semidirect
35. #16812609C22semidirect
36. C8×Dic109C22semidirect
62. #13182013D4semidirect
86. M16×C109D4semidirect
151. C109#3153semidirect
155. C2×AGL(1,ℤ/8)D109semidirect
228. M16C22⋊D109semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168127322
C2×#168127292
C2×#168127252
#202955722
C2×#168126862
#202954172
#202954142
#202954092
#202954072
M16×Dic1092
C2×AGL(1,ℤ/8)×C1092
#3153109